b How many points were used in Part A to find the equation? Are two points enough?
A
a (3,5)
B
bMinimum Number of Data Points: 3 Explanation: Substitute x- and y-values in the general form of a quadratic, the solve the linear system for the coefficients.
y = ax^2 + bx + c
In the equation above, the constant c represents the y-intercept. We are told that the points (0,-4), (2,4), and (4,4) lie on the parabola. Since (0,-4) corresponds to the y-intercept, we know that c=-4.
y = ax^2 + bx + (- 4)
⇕
y = ax^2 + bx - 4
To find the values of a and b, we will substitute the last two points in the equation above, and simplify.
y=ax^2+bx-4
Point
Substitute
Simplify
( 2, 4)
4=a( 2)^2+b( 2)-4
4a+2b=8
( 4, 4)
4=a( 4)^2+b( 4)-4
16a+4b=8
We can use the obtained equations to write a system of equations.
4a + 2b = 8 & (I) 16a + 4b = 8 & (II)
Let's solve the system by using the Elimination Method. To do so, we can multiply Equation (I) by 2 and subtract it from Equation (II).
Knowing that a= - 1, b= 6, and that c= - 4, we can write the equation of the parabola.
y= - 1x^2+ 6x+( - 4)
⇕
y = - x^2 + 6x - 4
Finally, we can find the x-coordinate of the vertex by using its corresponding formula.
x = -b/2a substitute ⟶ x = -6/2( -1) = 3
To find the y-coordinate, we substitute x=3 into the equation of the parabola.
b To find a quadratic equation that models a certain data set, we first consider the standard form of a quadratic function.
y = ax^2 + bx + c
As we can see, in the equation above there are three coefficients we must find, which are a, b, and c. This implies that we need at least three equations to form a system and solve it. To write three equations, we need three points.
